The Via Feratta or the regular route on the Alpspitze outside of Garmisch-Partenkirchen is fun, with great views. Snow conditions will dictate how easy it would be. May/June is fairly early in the season.

Moni's right - May / June is pretty early. For example, right now, in June almost everywhere in the Alps we still have snow above 2000m. Last year - after a hot period of two weeks in May - there was no snow left north of the Alps at that time.

Generally - you'll find all sorts of mountains over here. Most have regular hiking trails to their tops so finding one to bag should be no problem at all. There are few remote places, where you need to consider access route logistics. Most of the time you can start directly from the nearest village.

Moni's suggestion of the Alpspitze is a good one. I would add the Mittenwalder Klettersteig, a ferrata which heads out into the Karwendel Range from Mittenwald. Then there is the Hindelanger Klettersteig across the summits of the Daumen Group in the Allgäu Alps - but this is crowded at all times of the year.

http://www.oase-alpin.de/englisch/wande ... uerung.htm

http://www.summitpost.org/area/range/15 ... -alps.html

Maybe a mountain a little bit higher, but easy to bag.

Hochfeiler, on the border Austria/Italia near the Brenner-Pass.

A very nice hut, a good elevation gain, no glaciers, easy to go solo.

http://www.summitpost.org/route/207581/hochfeiler-southwest-ridge-normal-route-.html

http://www.summitpost.org/hut/campground/258613/hochfeilerhuette.html